Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A semi-analytical and numerical approach for solving 3D nonlinear cylindrical shell systems

  • Liming Dai (Sino-Canada Research Centre of Computation and Mathematics, Qinghai Normal University and the University of Regina, Qinghai Normal University) ;
  • Kamran Foroutan (Sino-Canada Research Centre of Computation and Mathematics, Qinghai Normal University and the University of Regina, Qinghai Normal University)
  • Received : 2022.09.25
  • Accepted : 2023.07.31
  • Published : 2023.09.10
⟨ Previous Next ⟩

Abstract

This study aims to solve for nonlinear cylindrical shell systems with a semi-analytical and numerical approach implementing the P-T method. The procedures and conditions for such a study are presented in practically solving and analyzing the cylindrical shell systems. An analytical model for a nonlinear thick cylindrical shell (TCS) is established on the basis of the stress function and Reddy's higher-order shear deformation theory (HSDT). According to Reddy's HSDT, Hooke's law in three dimensions, and the von-Kármán equation, the stress-strain relations are developed for the thick cylindrical shell systems, and the three coupled nonlinear governing equations are thus established and discretized as per the Galerkin method, for implementing the P-T method. The solution generated with the approach is continuous everywhere in the entire time domain considered. The approach proposed can also be used to numerically solve and analyze the nonlinear shell systems. The procedures and recurrence relations for numerical solutions of shell systems are presented. To demonstrate the application of the approach in numerically solving for nonlinear cylindrical shell systems, a specific nonlinear cylindrical shell system subjected to an external excitation is solved numerically. In numerically solving for the system, the present approach shows higher efficiency, accuracy, and reliability in comparison with that of the Runge-Kutta method. The approach with the P-T method presented is practically sound especially when continuous and high-quality numerical solutions for the shell systems are considered.

Keywords

Acknowledgement

The authors greatly appreciate the supports of the Natural Sciences and Engineering Research Council of Canada (NSERC), Qinghai Normal University and the University of Regina to the present research.

References

  1. Alijani, F., Amabili, M., Karagiozis, K. and Bakhtiari-Nejad, F. (2011), "Nonlinear vibration of functionally graded doubly curved shallow shells", J. Sound Vib., 330(7), 1432-1454. https://doi.org/10.1016/j.jsv.2010.10.003.
  2. Abukhaled, M.I. and Allen, E.J. (1998), "A class of second-order Runge-Kutta methods for numerical solution of stochastic differential equations", Stoch. Anal. Appl., 16(3), 977-992. https://doi.org/10.1080/07362999808809575.
  3. Chorfi, S.M. and Houmat, A. (2012), "Nonlinear free vibration of a functionally graded doubly curved shallow shell of elliptical planform", Compos. Struct., 92(10), 2573-2581. https://doi.org/10.1016/j.compstruct.2010.02.001.
  4. Dai, L. and Singh, M.C. (1997a), "An analytical and numerical method for solving linear and nonlinear vibration problems", Int. J. Solid. Struct., 34(21), 2709-2731. https://doi.org/10.1016/S0020-7683(96)00169-2.
  5. Dai, L. and Singh, M.C. (2003), "A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems", J. Sound Vib., 263(3), 535-548. https://doi.org/10.1016/S0022-460X(02)01065-9.
  6. Dai, L. and Singh, M.C. (1997b), "Diagnosis of periodic and chaotic responses in vibratory systems", J. Acoust. Soc. Am., 102(6), 3361-3371. https://doi.org/10.1121/1.420393.
  7. Dai, L., Chen, C. and Sun, L. (2015a), "An active control strategy for vibration control of an axially translating beam", J. Vib. Control, 21(6), 1188-1200. https://doi.org/10.1177/1077546313493312.
  8. Dai, L. (2008), Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific Publishing Co., New Jersey.
  9. Dai, L., Wang, X. and Chen, C. (2015b), "Accuracy and reliability of piecewise-constant method in studying the responses of nonlinear dynamic systems", J. Comput. Nonlin. Dyn., 10(2), 021009. https://doi.org/10.1115/1.4026895.
  10. Dai, L. and Wang, L. (2020), "Nonlinear analysis of high accuracy and reliability in traffic flow prediction", Nonlin. Eng., 9(1), 290-298. https://doi.org/10.1515/nleng-2020-0016.
  11. Dai, L., Xia, D. and Chen, C. (2019), "An algorithm for diagnosing nonlinear characteristics of dynamic systems with the integrated periodicity ratio and lyapunov exponent methods", Commun. Nonlin. Sci. Numer. Simul., 73, 92-109. https://doi.org/10.1016/j.cnsns.2019.01.029.
  12. Dai, L., Xu, L. and Han, Q. (1997), "Semi-analytical and numerical solutions of multi-degree-of-freedom nonlinear oscillation systems with linear coupling", Commun. Nonlin. Sci. Numer. Simul., 7(11), 831-844. https://doi.org/10.1016/j.cnsns.2004.12.009.
  13. Duc, N.D., Hadavinia, H., Quan, T.Q. and Khoa, N.D. (2019), "Free vibration and nonlinear dynamic response of imperfect nanocomposite FG-CNTRC double curved shallow shells in thermal environment", Eur. J. Mech. A Solid., 75, 355-366. https://doi.org/10.1016/j.euromechsol.2019.01.024.
  14. Ebrahimi, F., Dabbagh, A. and Rastgoo, A. (2023), "Static stability analysis of multi-scale hybrid agglomerated nanocomposite shells", Mech. Bas. Des. Struct. Mach., 51(1), 501-517. https://doi.org/10.1080/15397734.2020.1848585.
  15. Enright, W.H. and Hayes, W.B. (2007), "Robust and reliable defect control for Runge-Kutta methods", ACM Trans. Math. Softw., 33(1), 1-19. https://doi.org/10.1145/1206040.1206041.
  16. Eyvazian, A., Shahsavari, D. and Karami, B. (2020), "On the dynamic of graphene reinforced nanocomposite cylindrical shells subjected to a moving harmonic load", Int. J. Eng. Sci., 154, 103339. https://doi.org/10.1016/j.ijengsci.2020.103339.
  17. Foroutan, K. and Ahmadi, H. (2020), "Combination resonances of imperfect SSFG cylindrical shells rested on viscoelastic foundations", Struct. Eng. Mech., 75(1), 87-100. https://doi.org/10.12989/sem.2020.75.1.087.
  18. Foroutan, K., Ahmadi, H. and Shariyat, M. (2020), "Asymmetric large deformation superharmonic and subharmonic resonances of spiral stiffened imperfect FG cylindrical shells resting on generalized nonlinear viscoelastic foundations", Int. J. Appl. Mech., 12(05), 2050052. https://doi.org/10.1142/S1758825120500520.
  19. Daemi, H. and Eipakchi, H. (2020), "Effect of different viscoelastic models on free vibrations of thick cylindrical shells through FSDT under various boundary conditions", Struct. Eng. Mech., 73(3), 319-330. https://doi.org/10.12989/sem.2020.73.3.319.
  20. Hull, T.E., Enright, W.H., Fellen, B.M. and Sedgwick, A.E. (1972), "Comparing numerical methods for ordinary differential equations", SIAM J. Numer. Anal., 9(4), 603-637. https://doi.org/10.1137/0709052.
  21. Farahani, H. and Barati, F. (2015), "Vibration of sumberged functionally graded cylindrical shell based on first order shear deformation theory using wave propagation method", Struct. Eng. Mech., 53(3), 575-587. https://doi.org/10.12989/sem.2015.53.3.575.
  22. Kuinian, L. and Antony, P. (2009), "A high precision direct integration scheme for nonlinear dynamic systems", J. Comput. Nonlin. Dyn., 4(4), 041008. https://doi.org/10.1115/1.3192129.
  23. Matsunaga, H. (2008), "Free vibration and stability of functionally graded shallow shell according to a 2-D higher order deformation theory", Compos. Struct., 84(2), 132-146. https://doi.org/10.1016/j.compstruct.2007.07.006.
  24. Nakamura, S. (1991), Applied Numerical Methods with Software, Prentice Hall, New Jersey.
  25. Ninh, D.G. and Hoang, V.N.V. (2022), "A new shell study for dynamical characteristics of nanocomposite shells with various complex profiles-Sinusoidal and cosine shells", Eng. Struct., 251, 113354. https://doi.org/10.1016/j.engstruct.2021.113354.
  26. Ninh, D.G. and Tien, N.D. (2019), "Investigation for electro-thermo-mechanical vibration of nanocomposite cylindrical shells with an internal fluid flow", Aerosp. Sci. Technol., 92, 501-519. https://doi.org/10.1016/j.ast.2019.06.023.
  27. Thakur, S.N., Chakraborty, S. and Ray, C. (2019), "Reliability analysis of laminated composite shells by response surface method based on HSDT", Struct. Eng. Mech., 72(2), 203-216. https://doi.org/10.12989/sem.2019.72.2.203.
  28. Reddy J.N. (2004), Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton.
  29. Foroutan, K., Shaterzadeh, A. and Ahmadi, H. (2018), "Nonlinear dynamic analysis of spiral stiffened functionally graded cylindrical shells with damping and nonlinear elastic foundation under axial compression", Struct. Eng. Mech., 66(3), 295-303. https://doi.org/10.12989/sem.2018.66.3.295.
  30. Shampine, L.F. and Watts, H.A. (1971), "Comparing Error Estimators for Runge-Kutta Methods", Math. Comput., 25(115), 445-455. https://doi.org/10.1090/S0025-5718-1971-0297138-9.
  31. Sobhani, E., Arbabian, A., Civalek, O . and Avcar, M. (2021), "The free vibration analysis of hybrid porous nanocomposite joined hemispherical-cylindrical-conical shells", Eng. Comput., 1-28. https://doi.org/10.1007/s00366-021-01453-0.
  32. Sofiyev, A.H. (2015), "Influences of shear stresses on the dynamic instability of exponentially graded sandwich cylindrical shells", Compos. B. Eng., 77, 349-362. https://doi.org/10.1016/j.compositesb.2015.03.040.
  33. Sofiyev, A.H., Hui, D., Haciyev, V.C., Erdem, H., Yuan, G.Q., Schnack, E. and Guldal, V. (2017), "The nonlinear vibration of orthotropic functionally graded cylindrical shells surrounded by an elastic foundation within first order shear deformation theory", Compos. B. Eng., 116, 170-185. https://doi.org/10.1016/j.compositesb.2017.02.006.
  34. Thang, P.T., Duc, N.D. and Nguyen-Thoi, T. (2017), "Thermomechanical buckling and post-buckling of cylindrical shell with functionally graded coatings and reinforced by stringers", Aerosp. Sci. Technol., 66, 392-401. https://doi.org/10.1016/j.ast.2017.03.023.
  35. Raminnea, M., Biglari, H. and Tahami, F.V. (2016), "Nonlinear higher order Reddy theory for temperature-dependent vibration and instability of embedded functionally graded pipes conveying fluid-nanoparticle mixture", Struct. Eng. Mech., 59(1), 153-186. https://doi.org/10.12989/sem.2016.59.1.153.
  36. Van Dung, D. and Dong, D.T. (2016), "Post-buckling analysis of functionally graded doubly curved shallow shells reinforced by FGM stiffeners with temperature-dependent material and stiffener properties based on TSDT", Mech. Res. Commun., 78, 28-41. https://doi.org/10.1016/j.mechrescom.2016.09.008.
  37. Van Dung, D. and Nam, V.H. (2014), "Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium", Eur. J. Mech. A/Solid., 46, 42-53. https://doi.org/10.1016/j.euromechsol.2014.02.008.
  38. Van Tung, H. and Duc, N.D. (2014), "Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions", Appl. Math. Model., 38(11-12), 2848-2866. https://doi.org/10.1016/j.apm.2013.11.015.
  39. Wang, J., Rodriguez, J. and Keribar, R. (2010), "Integration of flexible multibody systems using Radau IIA algorithms", J. Comput. Nonlin. Dyn., 5(4), 041008. https://doi.org/10.1115/1.4001907.
  40. Zhang, L.W. and Selim, B.A. (2017), "Vibration analysis of CNT-reinforced thick laminated composite plates based on Reddy's higher-order shear deformation theory", Compos. Struct., 160, 689-705. https://doi.org/10.1016/j.compstruct.2016.10.102.
  41. Zhang, S. and Li, J. (2011), "Explicit numerical methods for solving stiff dynamical systems", J. Comput. Nonlin. Dyn., 6(4), 041008. https://doi.org/10.1115/1.4003706.
  42. Zingg, D.W. and Chisholm, T.T. (1999), "Runge-Kutta methods for linear ordinary differential equations", Appl. Numer. Math., 31(2), 227-238. https://doi.org/10.1016/S0168-9274(98)00129-9.